Q100 / 160 · TS EAMCET 2017 · Mathematics

If $\tan \theta_1=k \cot \theta_2$, then $\frac{\cos \left(\theta_1+\theta_2\right)}{\cos \left(\theta_1-\theta_2\right)}=$

  • A
    $\frac{1+k}{1-k}$
  • Correct answer
    B
    $\frac{1-k}{1+k}$
  • C
    $\frac{k+1}{k-1}$
  • D
    $\frac{k-1}{k+1}$

Solution

We have, $\tan \theta_1=k \cot \theta_2$ $\Rightarrow \quad \tan \theta_1 \tan \theta_2=k$ Consider, $\frac{\cos \left(\theta_1+\theta_2\right)}{\cos \left(\theta_1-\theta_2\right)}=\frac{\cos \theta_1 \cos \theta_2-\sin \theta_1 \sin \theta_2}{\cos \theta_1 \cos \theta_2+\sin \theta_1 \sin \theta_2}$ $$ =\frac{1-\tan \theta_1 \tan \theta_2}{1+\tan \theta_1 \tan \theta_2} $$ $$ =\frac{1-k}{1+k} $$

Hence, the correct answer is option: 'B'.

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